6 research outputs found

    A Tight Runtime Analysis for the cGA on Jump Functions---EDAs Can Cross Fitness Valleys at No Extra Cost

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    We prove that the compact genetic algorithm (cGA) with hypothetical population size μ=Ω(nlogn)poly(n)\mu = \Omega(\sqrt n \log n) \cap \text{poly}(n) with high probability finds the optimum of any nn-dimensional jump function with jump size k<120lnnk < \frac 1 {20} \ln n in O(μn)O(\mu \sqrt n) iterations. Since it is known that the cGA with high probability needs at least Ω(μn+nlogn)\Omega(\mu \sqrt n + n \log n) iterations to optimize the unimodal OneMax function, our result shows that the cGA in contrast to most classic evolutionary algorithms here is able to cross moderate-sized valleys of low fitness at no extra cost. Our runtime guarantee improves over the recent upper bound O(μn1.5logn)O(\mu n^{1.5} \log n) valid for μ=Ω(n3.5+ε)\mu = \Omega(n^{3.5+\varepsilon}) of Hasen\"ohrl and Sutton (GECCO 2018). For the best choice of the hypothetical population size, this result gives a runtime guarantee of O(n5+ε)O(n^{5+\varepsilon}), whereas ours gives O(nlogn)O(n \log n). We also provide a simple general method based on parallel runs that, under mild conditions, (i)~overcomes the need to specify a suitable population size, but gives a performance close to the one stemming from the best-possible population size, and (ii)~transforms EDAs with high-probability performance guarantees into EDAs with similar bounds on the expected runtime.Comment: 25 pages, full version of a paper to appear at GECCO 201

    From Understanding Genetic Drift to a Smart-Restart Parameter-less Compact Genetic Algorithm

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    One of the key difficulties in using estimation-of-distribution algorithms is choosing the population size(s) appropriately: Too small values lead to genetic drift, which can cause enormous difficulties. In the regime with no genetic drift, however, often the runtime is roughly proportional to the population size, which renders large population sizes inefficient. Based on a recent quantitative analysis which population sizes lead to genetic drift, we propose a parameter-less version of the compact genetic algorithm that automatically finds a suitable population size without spending too much time in situations unfavorable due to genetic drift. We prove a mathematical runtime guarantee for this algorithm and conduct an extensive experimental analysis on four classic benchmark problems both without and with additive centered Gaussian posterior noise. The former shows that under a natural assumption, our algorithm has a performance very similar to the one obtainable from the best problem-specific population size. The latter confirms that missing the right population size in the original cGA can be detrimental and that previous theory-based suggestions for the population size can be far away from the right values; it also shows that our algorithm as well as a previously proposed parameter-less variant of the cGA based on parallel runs avoid such pitfalls. Comparing the two parameter-less approaches, ours profits from its ability to abort runs which are likely to be stuck in a genetic drift situation.Comment: 4 figures. Extended version of a paper appearing at GECCO 202

    From Understanding Genetic Drift to a Smart-Restart Mechanism for Estimation-of-Distribution Algorithms

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    Estimation-of-distribution algorithms (EDAs) are optimization algorithms that learn a distribution on the search space from which good solutions can be sampled easily. A key parameter of most EDAs is the sample size (population size). If the population size is too small, the update of the probabilistic model builds on few samples, leading to the undesired effect of genetic drift. Too large population sizes avoid genetic drift, but slow down the process. Building on a recent quantitative analysis of how the population size leads to genetic drift, we design a smart-restart mechanism for EDAs. By stopping runs when the risk for genetic drift is high, it automatically runs the EDA in good parameter regimes. Via a mathematical runtime analysis, we prove a general performance guarantee for this smart-restart scheme. This in particular shows that in many situations where the optimal (problem-specific) parameter values are known, the restart scheme automatically finds these, leading to the asymptotically optimal performance. We also conduct an extensive experimental analysis. On four classic benchmark problems, we clearly observe the critical influence of the population size on the performance, and we find that the smart-restart scheme leads to a performance close to the one obtainable with optimal parameter values. Our results also show that previous theory-based suggestions for the optimal population size can be far from the optimal ones, leading to a performance clearly inferior to the one obtained via the smart-restart scheme. We also conduct experiments with PBIL (cross-entropy algorithm) on two combinatorial optimization problems from the literature, the max-cut problem and the bipartition problem. Again, we observe that the smart-restart mechanism finds much better values for the population size than those suggested in the literature, leading to a much better performance.Comment: Accepted for publication in "Journal of Machine Learning Research". Extended version of our GECCO 2020 paper. This article supersedes arXiv:2004.0714

    On the limitations of the univariate marginal distribution algorithm to deception and where bivariate EDAs might help

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    We introduce a new benchmark problem called Deceptive Leading Blocks (DLB) to rigorously study the runtime of the Univariate Marginal Distribution Algorithm (UMDA) in the presence of epistasis and deception. We show that simple Evolutionary Algorithms (EAs) outperform the UMDA unless the selective pressure μ/λ\mu/\lambda is extremely high, where μ\mu and λ\lambda are the parent and offspring population sizes, respectively. More precisely, we show that the UMDA with a parent population size of μ=Ω(logn)\mu=\Omega(\log n) has an expected runtime of eΩ(μ)e^{\Omega(\mu)} on the DLB problem assuming any selective pressure μλ141000\frac{\mu}{\lambda} \geq \frac{14}{1000}, as opposed to the expected runtime of O(nλlogλ+n3)\mathcal{O}(n\lambda\log \lambda+n^3) for the non-elitist (μ,λ) EA(\mu,\lambda)~\text{EA} with μ/λ1/e\mu/\lambda\leq 1/e. These results illustrate inherent limitations of univariate EDAs against deception and epistasis, which are common characteristics of real-world problems. In contrast, empirical evidence reveals the efficiency of the bi-variate MIMIC algorithm on the DLB problem. Our results suggest that one should consider EDAs with more complex probabilistic models when optimising problems with some degree of epistasis and deception.Comment: To appear in the 15th ACM/SIGEVO Workshop on Foundations of Genetic Algorithms (FOGA XV), Potsdam, German

    Self-Adjusting Evolutionary Algorithms for Multimodal Optimization

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    Recent theoretical research has shown that self-adjusting and self-adaptive mechanisms can provably outperform static settings in evolutionary algorithms for binary search spaces. However, the vast majority of these studies focuses on unimodal functions which do not require the algorithm to flip several bits simultaneously to make progress. In fact, existing self-adjusting algorithms are not designed to detect local optima and do not have any obvious benefit to cross large Hamming gaps. We suggest a mechanism called stagnation detection that can be added as a module to existing evolutionary algorithms (both with and without prior self-adjusting algorithms). Added to a simple (1+1) EA, we prove an expected runtime on the well-known Jump benchmark that corresponds to an asymptotically optimal parameter setting and outperforms other mechanisms for multimodal optimization like heavy-tailed mutation. We also investigate the module in the context of a self-adjusting (1+λ\lambda) EA and show that it combines the previous benefits of this algorithm on unimodal problems with more efficient multimodal optimization. To explore the limitations of the approach, we additionally present an example where both self-adjusting mechanisms, including stagnation detection, do not help to find a beneficial setting of the mutation rate. Finally, we investigate our module for stagnation detection experimentally.Comment: 26 pages. Full version of a paper appearing at GECCO 202
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